Formulae Geom 1

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Geometry Formulae


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Basic Geometry


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Basic Geometry

If two or more angles forma straight angle, the sum oftheir measures is 180

file:///Z:/Student%20Facilitation/Lessons%20and%20Tutorials/SFA%20Zone/Sunitha/formulae%20bank/geometry/line-angle.png

The sum of all the measures of allthe angles around a point is 360


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Parallel Lines

If a pair of straight lines is cut by atransversal that is not perpendicularto the parallel lines, then

Vertically opposite angles are a=b,

c=d, e= f, g=h Corresponding angles are a=e,

c= g, d=h, b=f

Alternate interior angles are c=h,

e=b Alternate exterior angles are a=f,

d=g

Supplementary angle pairs are c+e= b+h = 180


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Triangles


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Triangle

In any triangle, the sum of the measures of the threeangles is 180

The measure of the exterior angle of the triangle is equalto the sum of the measure of the two opposite interior

angles In any right triangle, the sum of the measures of the two

acute angles is 90


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30

60

90

A

B C

In most GRE geometry problems, you need not knowTrigonometry.A few equivalent concepts can help you solve them.

In many problems, the triangles turnout to be a right triangle with one of the angles as 30 or 60.You have to immediately register that this is a

30-60-90 Triangleand check if there is an opportunityfor you to apply the following rule.This rule helps you to determinelength of 2 sides of the Triangle, if you know just one

The rule says that: In a 30-60-90 triangle (one shown above),The ratio of the

The length of side opposite to 30 : The length of side opposite to 60 :The length of side opposite to 90

= 1 : 3 : 2 = AB : BC : AC

.

30-60-90 Triangle


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Sometimes, the triangles turn out to be a isosceles righttriangle with one of the angles as 45.You have to immediately register that this is a45-45-90 Triangleand check if there is an opportunityfor you to apply the following rule.This rule helps you to determinelength of the sides of the Triangle, if you know just one

A

B C

45

4590

The rule says that: In a 45-45-90 triangle(shown triangle),

The ratio of theThe length of side opposite to 45 : The lengthof side opposite to 45 : The lengthof side opposite to 90

= 1 : 1:2 = AB : BC : AC

45-45-90 Triangle


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Triangle

An altitude divides an equilateral triangle into two 30-60-90triangles

The sum of the lengths of any two sides of a triangle is greaterthan the length of the third side

The difference of the lengths of any two sides of a triangle is lessthan the length of the third side

The area of the triangle is given byA= b.h, where b is the baseof the triangle and h is the height of the triangle

The area of an equilateraltriangle with side s is given by

A = s3/4


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Centroid

Centroid is the meeting point of the mediansdrawn from the vertex to the mid-point of theopposite side of the triangle

Centroid divides the median in the ratio 1 : 2

Thus in the adjoining figure GE/BE = 1/3

or GE/BG = 1/2

Similarly, GD/AD = 1/3

or GD/AG = 1/2


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Circles


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Circles

Any triangle formed byconnecting the endpoints oftwo radii, is isosceles. HereOPQ = OQP

Circumference = 2r = d,where d is the diameter ofthe circle

Area = r

Circumference of a semicircle = r + d


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Arc of a circle

Degree measure of a complete circle is 360 The degree measure of an arc AB = x

O

A

B

xx


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Arcs on a circle

Length of a Arc AB / Circumference = x/360 Area of a sector AOB / Area of the Circle = x/360

O

A

B

xxx


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Angle Properties

The angle subtended by a chord atthe centre of the circle is twice theangle subtended by the Chord AB

on the circle. Note that the angles (in violet) onthe arc AQB are all equal angles

If AOB = 2x, then APB = x;

AQB = x; ARB = xA B

O

P

Q

R


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Coordinate Geometry

The distance, d, between two given points, A(x1, y1) and B(x2,y2), can be calculated using the distance formula, d = (x2-x1)+ (y2- y1)

Vertical lines do not have slopes

Slope of any horizontal line is 0 Slope of a line when two points are given is

m= (y 2-y1)/(x2-x1)


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Equation of a line

Equation of line: y = mx + b (y y1) / (x x1) = m, if (x1,y1) is a point on the line

x/x0

+ y/y0

= 1, if (0, y0) and (x

0, 0) are the intercepts

(0, y0)

(x0, 0)

X (x1, y1)


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Equation of a line

Equation of a line parallel to X axis is : y = y0(where y0 is the Yco-ordinate of the point where the line intersects the Y axis)

Equation of a line parallel to Y axis is : x = x0

(where x0

is the X

co-ordinate of the point where the line intersects the X axis)

(0, y0)

(x0

, 0)

X

X (0,0) X

Y


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Polygons


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Polygon types

Name Number of Sides

Triangle 3

Quadrilateral 4

Pentagon 5Hexagon 6

Heptagon 7

Octagon 8

9

Decagon 10

Nanogon


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Cyclic Polygon

A convex Polygon is called a Cyclic Polygon, ifall the vertices lie on a single circle

Sum of opposite angles of a Cyclic

Quadrilateralis 180 a

d

b

c

a + c = 180

b + d = 180

A Regular polygon is a Cyclic Polygon whosesides are of Equal length


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Formulae related to Polygons

Sum ofInteriorAngles of a N sided Polygon =(N 2) x 180

The interior angles of a Regular Polygon are

equal to each other. The measure of an interiorangle of a regular Polygon =

(N 2) x 180 / N

Number of Diagonals of a N sides polygon =N x (N-3) / 2


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Angles Sum ofExternal Angles of a N sided Polygon =

(n+2) * 180 The measure of each ExteriorAngle is 360 / N The External angle is different and marked in

blue for reference


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Parallelogram

AB = DC and AD = BC a = c , b = d

a + b = 180

c + d = 180

b+c = 180 & a+d = 180

Diagonals AC and BD bisect

each other A diagonal divides the

parallelogram into twoCongruent triangles

a b

d c

A


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Rectangle

AB = DC and AD = BC Angles a = c = b = d = 90

Diagonals AC and BDbisect each other

The diagonals of a rectangle have the same length, AC = BD

a b

d c

B

D


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Square AB = DC = AD = BC a = c = b = d = 90 Diagonals AC and BD bisect

each other at right angles andare perpendicular to

each other

AEB, BEC, CED, DEAare 454590 triangles

a b

d c

A

CD

E

9045 45


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Area

Parallelogram : DC x h1

= BC x h2

Rectangle : length x breadth Square : (side)2 = (diagonal)2/ 2

Trapezium :() (b1 + b2) x h

A

D

h1

h2

h

b2

b1


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Some points to remember

For a given Perimeter, the rectangle with thelargest area is a square.

For a given area , the rectangle with the

smallest perimeteris a square


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Solid Geometry


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Cuboid

The volume of a rectangular solid (cuboid), isV= l.w.h

ThesurfaceareaofacuboidisA=2(lw + lh +wh)

l

b

w


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Cube

The volume of the cube is V = a.a.a = a The surface area is A = 6.a

a a

a


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Diagonal

A diagonal,d,of a box is the longest linesegment that can be drawn between two pointson the box, d = l + w + h


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Cylinder

The Volume of a Cylinder, V whose circularbase has radius r and height h is V = rh

The surface area, A, of the cylinder isA = 2rh

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Formulae Geom 1

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